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Solar radiation model

L.T. Wong, W.K. Chow *

Department of Building Services Engineering, The Hong Kong Polytechnic University,

Hung Hom, Kowloon, Hong Kong, China

Abstract

Solar radiation models for predicting the average daily and hourly global radiation, beam

radiation and diffuse radiation are reviewed in this paper. Seven models using the A ngstro m

Prescott equation to predict the average daily global radiation with hours of sunshine are

considered. The average daily global radiation for Hong Kong (22.3N latitude, 114.3E

longitude) is predicted. Estimations of monthly average hourly global radiation are discussed.

Two parametric models are reviewed and used to predict the hourly irradiance of Hong Kong.Comparisons among model predictions with measured data are made. # 2001 Elsevier

Science Ltd. All rights reserved.

1. Introduction

A knowledge of the local solar-radiation is essential for the proper design of

building energy systems, solar energy systems and a good evaluation of thermal

environment within buildings [16]. The best database would be the long-termmeasured data at the site of the proposed solar system. However, the limited cover-

age of radiation measuring networks dictates the need for developing solar radiation

models.

Since the beam component (direct irradiance) is important in designing systems

employing solar energy, such as high-temperature heat engines and high-intensity

solar cells, emphasis is often put on modelling the beam component. There are two

categories of solar radiation models, available in the literature, that predict the beam

component or sky component based on other more readily measured quantities:

Applied Energy 69 (2001) 191224

www.elsevier.com/locate/apenergy

0306-2619/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved.

P I I : S 0 3 0 6 - 2 6 1 9 ( 0 1 ) 0 0 0 1 2 - 5

* Corresponding author. Tel.: +852-2766-5111; fax: +852-2765-7198.

E-mail address:[email protected] (W.K. Chow).

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Nomenclature

A Apparent extraterrestrial irradiance (W m

2)a1, a2,a3 Coefficients defined in Eqs. (33), (50) and (68), respectively

B Overall broadband value of the atmospheric attenuation-

coefficient for the basic atmosphere (dimensionless)

b1, b2,b3 Coefficients defined in Eqs. (33), (50) and (68), respectively

C Ratio of the diffuse radiation falling on a horizontal sur-

face under a cloudless sky Id to the direct normal irradia-

tion at the Earths surface on a clear day In(dimensionless)

Cn Clearness number, the ratio of the direct normal irradiance

calculated with local mean clear-day water vapour divided

by the direct normal irradiance calculated with watervapour according to the basic atmosphere (dimensionless)

c1; c2; c3; c03; c4; c

04 Coefficients defined in Eqs. (81)(92)

D:a Aerosols scattered diffuse irradiance after the first pass

through the atmosphere (W m2)

D:m Broadband diffuse irradiance on the ground due to the

multiple reflections between the Earths surface and its

atmosphere (W m2)

D:r Rayleigh scattered diffuse irradiance after the first pass

through the atmosphere (W m2)

d1; d2; d3; d4; d5; d6 Coefficients defined in Eqs. (81)(92)

Eo Eccentricity correction factor of the Earths orbit (dimen-

sionless)

Fc Fraction of forward scattering to total scattering (dimen-

sionless)

Fsg Angle factor between the surface and the Earth (dimen-

sionless)

Fss Angle factor between the tilt surface and the sky (dimen-

sionless)

Gb Monthly average daily beam irradiance on a horizontalsurface (MJ m2)

Gc Monthly average daily clear-sky global radiation incident

on a horizontal surface (MJ m2)

Go Monthly average daily extraterrestrial radiation on a hor-

izontal surface (MJ m2)

Gt Monthly average daily global radiation on a horizontal

surface (MJ m2)

h Elevation above sea level (km)

I:b; Ib Total beam irradiance on a horizontal surface (W m

2);

hourly beam irradiance on a horizontal surface (MJ m

2)I:d; Id Total diffuse irradiance on a horizontal surface (W m

2);

192 L.T. Wong, W.K. Chow / Applied Energy 69 (2001) 191224

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hourly diffuse irradiance on a horizontal surface (MJ

m2)

I:n; In Direct normal irradiance (W m

2); hourly normal irra-diance (MJ m2)

I:o; Io Extraterrestrial radiation (W m

2); hourly extraterrestrial

radiation (MJ m2)

I:r Reflected short-wave radiation (W m

2)

I:sc; Isc Solar constant, 1367 W m

2; solar constant in energy unit

for 1 h (13673.6 kJ m2)

I:t; It Global solar irradiance (W m

2), hourly global solar irra-

diance (MJ m2)

kb Atmospheric transmittance for beam radiation (dimen-

sionless)kD Atmospheric diffuse coefficient (dimensionless)

kd Diffuse fraction (dimensionless)

kt Clearness index (dimensionless)

lao Aerosol optical-thickness (dimensionless)

loz Vertical ozone layer thickness (cm)

lRo Rayleigh optical-thickness (dimensionless)

ma Air mass at actual pressure (dimensionless)

mr Air mass at standard pressure of 1013.25 mbar (dimension-

less)

N Day number in the year (No.)

p Local air pressure (mbar)

po Standard pressure (1013.25 mbar)

qrat Downward radiation from the atmosphere due to long-

wave radiation (W m2)

RH Monthly average daily relative humidity (%)

rt Ratio defined in Eq. (53) (dimensionless)

S Monthly average daily sunshine-duration (h)

Sf Monthly average daily fraction of possible sunshine,

known as sunshine fraction (dimensionless)So Maximum possible monthly average daily sunshine-duration

(h)

Som Modified average daily sunshine duration for solar zenith

angle z greater than 85 (h)

T Monthly average daily-temperature (C)

Tat Atmosphere temperature at ground level (K)

Tdew Dew-point temperature (C)

TL Linke turbidity-factor defined in Eq. (112)

Tsky Apparent sky-temperature (K)

U1 Pressure-corrected relative optical-path length of pre-cipitable water (cm)

L.T. Wong, W.K. Chow / Applied Energy 69 (2001) 191224 193

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U3 Ozone relative optical path length under the normal

temperature and surface pressure NTP (cm)Vis Horizontal visibility (km)

w Precipitable water-vapour thickness reduced to the stan-

dard pressure of 1013.25 mbar and at the temperature T of

273K (cm)

w0 Precipitable water-vapour thickness under the actual con-

ditions (cm)

Change of

Day angle (in radians)

Solar altitude (degrees)

1; 2 A ngstro m turbidity-parameters (dimensionless) Declination, north positive (degrees)

c Declination of characteristics days, north positive (degrees)

"at Atmosphere emittance for long-wave radiation (dimen-

sionless)

Latitude, north positive (degrees)

g Reflectance of the foreground (dimensionless)

l Wavelength (mm)

Angle of incidence; the angle between normal to the sur-

face and the SunEarth line (degrees)

z Zenith angle (degrees)

a Albedo of the cloudless sky (dimensionless)

c Albedo of cloud (dimensionless)

g Albedo of ground (dimensionless)

Stefan-Boltzmann constant (5.67108 W m2 K4)

r; o; g; w; a Scattering transmittances for Rayleigh, ozone, gas, water

and aerosols (dimensionless)

aa Transmittance of direct radiation due to aerosol absorp-

tance (dimensionless)

as Transmittance of direct radiation due to aerosol scattering(dimensionless)

! Hour angle, solar noon being zero and the morning is

positive (degrees)

!I Hour angle at the middle of an hour (degrees)

!o Single-scattering albedo (dimensionless)

!s Sunset-hour angle for a horizontal surface (degrees)

Function

Tilt angle of a surface measured from the horizontal

(degrees)

< > Yearly average


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. Parametric models

. Decomposition models

Parametric models require detailed information of atmospheric conditions.Meteorological parameters frequently used as predictors include the type, amount,

and distribution of clouds or other observations, such as the fractional sunshine,

atmospheric turbidity and precipitable water content [5,714]. A simpler method

was adopted by the ASHRAE algorithm [5] and very widely used by the engineering

and architectural communities. The Iqbal model [8] offers extra-accuracy over more

conventional models as reviewed by Gueymard [13].

Development of correlation models that predict the beam or sky radiation using

other solar radiation measurements is possible. Decomposition models usually use

information only on global radiation to predict the beam and sky components.

These relationships are usually expressed in terms of the irradiations which are the

time integrals (usually over 1 h) of the radiant flux or irradiance. Decomposition

models developed to estimate direct and diffuse irradiance from global irradiance

data were found in the literature [1523].

Solar radiation models commonly used for systems using solar energy are

reviewed in this article. Comparisons of predicted values with these models and

measured data in Hong Kong are made. The units and symbols follow those sug-

gested by Beckman et al. [24].

2. Parametric models

2.1. Iqbal model C [8]

The direct normal irradiance I:n(W m

2) in model C described by Iqbal [8] is given by

I:n 0:9751EoI

:scrogwa 1

where the factor 0.9751 is included because the spectral interval considered is 0.33

mm; I

:

sc is the solar constant which can be taken as 1367 W m2

. Eo (dimensionless)is the eccentricity correction-factor of the Earths orbit and is given by

Eo 1:000110:034221cos 0:00128sin0:000719cos2 0:000077sin2

2

where the day angle (radians) is given by

2N1

365

3

where N is the day number of the year, ranging from 1 on 1 January to 365 on 31

December.


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r, o, g, w, a (dimensionless) are the Rayleigh, ozone, gas, water and aerosols

scattering-transmittances respectively. They are given by

r e0:0903m0:84a 1mam

1:01a 4

o 1

0:1611U3 1139:48U3 0:30350:002715U3 10:044U30:0003U

23

1h i5

g e0:0127m0:26a 6

w 1 2:4959U1 179:034U1 0:68286:385U1

17

a el0:873ao 1loal

0:7808ao m0:9108a 8

wherema (dimensionless) is the air mass at actual pressure andmr (dimensionless) is

the air mass at standard pressure (1013.25 mbar). They are related by

ma mrp

1013:25

9

where p (mbar) is the local air-pressure.

U3 (cm) is the ozones relative optical-path length under the normal temperature

and surface pressure (NTP) and is given by

U3 lozmr 10

where loz (cm) is the vertical ozone-layer thickness.

U1(cm) is the pressure-corrected relative optical-path length of precipitable water,as given by

U1 wmr 11

where w (cm) is the precipitable water-vapour thickness reduced to the standard

pressure (1013.25 mbar) and at the temperature Tof 273 K: w is calculated with the

precipitable water-vapour thickness under the actual condition w0 (cm) by

w w0 p

1013:25

34 273

T

12

12


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lao(dimensionless) is the aerosol optical thickness given by

lao 0:2758lao;ljl0:38 mm 0:35lao;ljl0:5 mm 13

where l (mm) is the wavelength.

The beam component I:b (W m

2) is given by

I:b coszI

:n 14

where z (degrees) is the zenith angle.

The horizontal diffuse irradiance I:d (W m

2) at ground level is a combination of

three individual components corresponding to the Rayleigh scattering after the first

pass through the atmosphere,D:r (W m

2); the aerosols scattering after the first pass

through the atmosphere,D: a (W m2); and the multiple-reflection processes between

the ground and sky D:m, (W m

2).

I:d D

:r D

:a D

:m

where

D:r

0:79I:scsinogwaa0:5 1r

1ma m1:02a16

where (degrees) is the solar altitude and is related to the zenith angle z by the

following equation

cosz sin 17

aa (dimensionless) is the transmittance of direct radiation due to aerosol absorp-

tance and is given by

aa 1 1!o 1mam1:06a

1a 18

where !o (dimensionless) is the single-scattering albedo fraction of incident energy

scattered to total attenuation by aerosols and is taken to be 0.9.

D:a

0:79I:scsinogwaaFc 1as

1mam1:02a19

Fc (dimensionless) is the fraction of forward scattering to total scattering and is

taken to be 0.84;as(dimensionless) is the fraction of the incident energy transmitted

after the scattering effects of the aerosols and is given by

as a

aa20


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D:m

I:nsinD

:rD

:a

ga

1ga21

where g (dimensionless) is the ground albedo; a (dimensionless) is the albedo of

the cloudless sky and can be computed from

a 0:0685 1Fc 1as 22

where (1 Fc) is the back-scatterance. Consequently, the second term on the right-

hand side of this equation represents the albedo of cloudless skies due to the pre-

sence of aerosols, whereas the first term represents the albedo of clean air.

The global (direct plus diffuse) irradiance I:t (W m

2) on a horizontal surface can

be written as

I:t I

:b I

:d I

:ncoszD

:r D

:a

11ga

23

2.2. ASHRAE model [5]

A simpler procedure for solar-radiation evaluation is adopted in ASHRAE [5] and

widely used by the engineering and architectural communities. The direct normal

irradiance I

:

n (W m

2

) is given by

I:n Cn Ae

Bsecz 24

whereA (W m2) is the apparent extraterrestrial irradiance, as given in Table 1, and

which takes into account the variations in the SunEarth distance. Therefore

Table 1

Values of A, B and C for the calculation of solar irradiance according to the ASHRAE handbook:

applications [5]

Date A (W m2) B(W m2) C(W m2)

21 Jan 1230 0.142 0.058

21 Feb 1215 0.144 0.060

21 Mar 1186 0.156 0.071

21 Apr 1136 0.180 0.097

21 May 1104 0.196 0.121

21 Jun 1088 0.205 0.134

21 Jul 1085 0.207 0.136

21 Aug 1107 0.201 0.122

21 Sep 1152 0.177 0.092

21 Oct 1193 0.160 0.07321 Nov 1221 0.149 0.063

21 Dec 1234 0.142 0.057


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A I:scEo constant 25

The value of the multiplying constant is close to 0.9 [8]. The variable B (dimen-

sionless) in Table 1 represents an overall broadband value of the atmosphericattenuation coefficient for the basic atmosphere of Threlkeld and Jordan [5]. Cn(dimensionless) is the clearness number and the map of Cn values for the USA is

provided in the ASHRAE handbook: applications [5]. Cn is the ratio of the direct

normal irradiance calculated with the local mean clear-day water-vapour to the

direct normal irradiance calculated with water vapour according to the basic atmo-

sphere.

Eq. (24) was developed for sea-level conditions. It can be adopted for other

atmospheric pressures by

I:n Cn Ae

Bcoszp

po

CnAe

B ppo

secz 26

where p (mbar) is the actual local-air pressure and po is the standard pressure

(1013.25 mbar).

In the above equation, the term (p=po) seczapproximates to the air mass, with the

assumptions that the curvature of the Earth and the refraction of air are negligible.

The total solar irradianceI:t (W m

2) of a terrestrial surface of any orientation and

tilt with an incident angle(degrees) is the sum of the direct component I:n cos plus

the diffuse component I

:

d coming from the sky, plus whatever amount of reflectedshort-wave radiation I:r (W m2) that may reach the surface from the Earth or from

adjacent surfaces, i.e.,

I:t I

:ncosI

:d I

:r 27

The diffuse component is defined through a variable C (dimensionless) given in

Table 1. The variable Cis the ratio of the diffuse radiation falling on a horizontal

surface under a cloudless sky I:d to the direct normal irradiation at the Earths sur-

face on a clear day I:n. The diffuse radiation I

:d is given by

I:d CI

:nFss 28

where for a horizontal surface,

CI:d

I:n

29

Fss (dimensionless), the angle factor between the surface and the sky, is given by

Fss

1 cos

2 30

where (degrees) is the tilt angle of a surface measured from the horizontal.


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The reflected radiation I:r (W m

2) from the foreground is given by

I:r I

:t;0gFsg 31

where g (dimensionless) is the reflectance of the foreground; and I:t;0 is the total

horizontal irradiation ( 0). Fsg, the angle factor between the surface and the

earth, is given by

Fsg 1 cos

2 32

3. Comparison of the Iqbal model C [8] and ASHRAE model [5]

The direct normal irradiance In was calculated with the Iqbal model C [8] for an

ozone thickness of 0.35 cm (NTP) at the 21st day of each month for various values

of the zenith angle z for Hong Kong (22.3N latitude, 114.3E longitude). The

beam, diffuse and global irradiances I:b, I

:d and I

:t respectively, were computed using

both models and are shown in Fig. 1(a)(c). The predicted values for January and

June are shown in Fig. 2(a)(c) for comparison.

At z 0, a maximum difference for the beam irradiance of not more than 7% is

obtained between the two models. The beam irradiance deviation of less than 8% was

found for SeptemberApril, and less than 12% for MayAugust when z

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Fig. 1. Comparison of Iqbal model and ASHRAE model.


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Fig. 2. Variation with zenith angle in January and June.


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cloud amount, thickness, position and the number of layers. These data are rarely

collected on a routine basis. However, sunshine hours and total cloud-cover data

(i.e. the fraction of sky covered by clouds) are widely and easily available. Correla-

tions, therefore, were developed to estimate insolation and sunshine hours.The monthly average daily global radiation on a horizontal surface can be esti-

mated through the number of bright sunshine hours. A ngstro m [25] developed the

first model suggesting that the ratio of the average daily global radiation Gt (MJ

m2) and cloudless radiation Gc (MJ m2) is related to the monthly mean daily

fraction of possible sunshine (sunshine fraction) Sf (dimensionless) by

Gt

Gca1 1a1 Sf a1 b1Sf 33

with the constant a1being obtained as 0.25 for Stockholm, Sweden.

The sunshine fraction Sfis obtained from:

Sf S

So34

where S (h) is the monthly average number of instrument-recorded bright daily

sunshine hours, and So (h) is the average day length. In the A ngstro m model,

a1b1 1, because on clear days, Sfis supposed to equal unity. However, becauseof problems inherent in the sunshine recorders, measurements of Sf would never

equal unity. This leads to an underestimation of the insolation.

A modified form of the above equation, known as the A ngstro mPrescott for-

mula, was reported by Prescott in 1940 [26]:

Gt

Gca1b1Sf 35

where a1 0:22 andb1 0:54.In Eqs. (33) and (35), a1 and b1 correspond respectively to the relative diffuse

radiation on an overcast day, whereas (a1 b1) corresponds to the relative cloud-

less-sky global irradiation. These equations assume that Gt corresponds to an idea-

lised day over the month and can be obtained as the weighted sum of the global

irradiations accumulated during two limiting days with extreme sky conditions, i.e.

fully overcast and perfectly cloudless.

There may be problems in calculating Gc accurately. This equation was modified

to be based on the extraterrestrial irradiation by Prescott in 1940 as reviewed by

Gueymard et al. [26]:

Gt

Goa1b1

S

So36


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In equation (36), Gt and Go (MJ m2) are the monthly mean daily global and

extraterrestrial radiations on a horizontal surface, a1 and b1 are constants for dif-

ferent locations.

The hourly extraterrestrial radiation on a horizontal surface Go (MJ m2) can bedetermined by [8]

Go 24

IscEocoscos sin!s

180

!scos!s

h i 37

where Isc is the solar constant (=13673.6 kJ m2 h1), and !s (degrees) is the

sunset-hour angle for a horizontal surface.

Rietveld [27] examined several published values ofa1and b1and noted that a1 is

related linearly and b1 hyperbolically to the appropriate yearly average value ofSf,

denoted as

SSo

, such that

a1 0:100:24 S

So

38

b1 0:380:08 So

S

39

If SSoD E

SSo

, the Rietveld model [27] can be simplified [26] to a constant-coefficient

A ngstro mPrescott equation by substituting Eqs. (38) and (39) into (36):

Gt

Go0:180:62

S

So40

This equation was proposed for all places in the world and yields particularly

superior results for cloudy conditions when SSo

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where (degrees) is the latitude, andh (km) is the elevation of the location above sea

level.

The coefficients a1 and b1 are site dependent [20,2932]. A pertinent summary is

shown in Table 2. The correlations of the average daily global radiation ratio andthe sunshine duration ratio in Hong Kong (22.3N latitude) are shown in Fig. 3 for

comparison. The monthly average daily global radiation ratio registered in 1961

1990 in Hong Kong [33] is shown. It is very close to the predictions of Bahel et al.

[29], Alnaser [30] and Louche et al. [20].

These coefficients would be affected by the optical properties of the cloud cover,

ground reflectivity, and average air mass. Hay [34] studied these factors with data

collected in western Canada and proposed a site-independent correlation

Gt

Go

0:15720:556

S

Som

1g aS

Som

c 1

S

Som

44

where g (dimensionless) is the ground albedo, a is the cloudless-sky albedo taken

as 0.25 andc is the cloud albedo taken as 0.6.Som(h) is the modified day-length for

a solar zenith angle z greater than 85 and is given by

Som 1

7:5

cos1 cos85sinsinc

coscosc 45

where c (degrees) is the characteristics declination, i.e. the declination at which the

extraterrestrial irradiation is identical to its monthly average value.

Sen [32] employed fuzzy modelling to represent the relations between solar irra-

diation and sunshine duration by a set of fuzzy rules. A fuzzy logic algorithm has

been devised for estimating the solar irradiation from sunshine duration measure-

ments. The classical A ngstro m correlations are replaced by a set of fuzzy-rule bases

Table 2Some examples of regression coefficients a1and b1used in different models

Reference a1 b1 Data source

[27] 0:1 0:24 SSo

D E 0:38 0:08 S

So

D E World-wide (42 places) (6 <

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and applied for three sites with monthly averages of daily irradiances in the western

part of Turkey.

Mohandes et al. [35] studied the monthly mean daily values of solar radiation

falling on horizontal surfaces using the radial basis functions technique. The pre-dicted results by a multilayer perceptrons network were compared with the simpli-

fied Rietveld model [27]. The predicted results by the neural network were compared

with those measured data for ten locations and good agreement was found.

5. Estimation of monthly-average hourly global radiation

As reviewed by Iqbal [8], Whiller investigated the ratios of hourly global radiation

It (MJ m2) to daily global radiation Gt(MJ m

2) in 1956 and assumed that

It

Gt

Io

Go46

Fig. 3. Correlation of average daily global radiation with sunshine duration for Hong Kong ( 23).


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where Io is the extraterrestrial hourly radiation (MJ m2), which can be deter-

mined by

Io IscEo sinsin0:9972coscoscos!I 47

whereIscis the solar constant, Eois the eccentricity correction factor, , and I are

the declination, latitude and hour angle respectively at the middle of an hour in

degrees.

Whiller [8] correlated the hourly and daily beam radiations Ib (MJ m2) and Gb

(MJ m2) respectively, as

Ib

Gb

Io

Go

1hkb!I

:scEocoszd!

1daykb!I

:

scEocoszd!

48

where ! (degrees) is the hour angle, i.e., zero for solar noon and positive for the

morning.

Assuming that the atmospheric transmittance for beam radiation kb ! to be

constant throughout the day, this equation can be rewritten as

Ib

Gb

Io

Go

24

24

sin

24

cos!Icos!s

sin!s

180

!scos!s

49

where !s is the sunset-hour angle (in degrees) for a horizontal surface.

Liu and Jordan [15] extended the day-length range of Whillers data and plotted

the ratio rt ItGt

against the sunset-hour angle !s. The mathematical expression was

presented by Collares-Pereira and Rabl [36]

rt It

Gt

Io

Goa2 b2cos!I 50

where

a2 0:4090:5016sin !s 60 51

b2 0:66090:4767sin !s 60 52

With Eqs. (37) and (47) for Io and Go, Eq. (50) can be expressed as

rt 24

cos!I cos!s

sin!s !s

180cos!s

0B@ 1CA a2b2cos!I 53


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A graphical presentation is shown in Fig. 4 and the graph is applicable everywhere

[e.g. 8,15,29].

6. Estimation of the hourly diffuse radiation on a horizontal surface using

decomposition models

Values of global and diffuse radiations for individual hours are essential for

research and engineering applications. Hourly global radiations on horizontal sur-

faces are available for many stations, but relatively few stations measure the hourly

diffuse radiation. Decomposition models have, therefore, been developed to predict

the diffuse radiation using the measured global data.

The models are based on the correlations between the clearness index kt (dimen-

sionless) and the diffuse fraction kd (dimensionless), diffuse coefficient kD (dimen-

sionless) or the direct transmittance kb (dimensionless) where

Fig. 4. Distribution of the monthly average hourly global radiation.


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kt It

Io54

kd Id

It55

kD Id

Io56

kb Ib

Io57

It, Ib, Id and Io being the global, direct, diffuse and extraterrestrial irradiances,

respectively, on a horizontal surface (all in MJ m2).

6.1. Liu and Jordan model [15]

The relationships permitting the determination, for a horizontal surface, of the

instantaneous intensity of diffuse radiation on clear days, the long-term average

hourly and daily sums of diffuse radiation, and the daily sums of diffuse radiation

for various categories of days of differing degrees of cloudiness, with data from 98

localities in the USA and Canada (19 to 55N latitude), were studied by Liu and

Jordan [15].The transmission coefficient for total radiation on a horizontal surface is given by

the intensity of total radiation (i.e. direct Ib plus diffuse Id) incident upon a hor-

izontal surface It divided by the intensity of solar radiation incident upon a hor-

izontal surface outside the atmosphere of the Earth Io. The correlation between the

intensities of direct and total radiations on clear days is given by

kD 0:2710:2939kb 58

Since

kt Ib ID =Io kb kD 59

then

kD 0:3840:416kt 60

6.2. Orgill and Hollands model [16]

Following the work by Liu and Jordan [15], the diffuse fraction was estimated byOrgill and Hollands [16] using the clearness indexktas the only variable. The model

was based on the global and diffuse irradiance values registered in Toronto (Canada,


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42.8N) in the years 19671971. The correlation between hourly diffuse fraction kdand clearness index ktis given by

kd 1 0:249kt; kt 0:75 63

Once the hourly diffuse fraction is obtained from the kdkt correlations, the

direct irradianceIb is obtained by [23]

Ib

Gt 1kd

sin 64

where is the solar elevation angle.

6.3. Erbs et al. model [17]

Erbs et al. [17] studied the same kind of correlations with data from 5 stations in

the USA with latitudes between 31 and 42. The data were of short duration, ran-

ging from 1 to 4 years. In each station, hourly values of normal direct irradiance and

global irradiance on a horizontal surface were registered. Diffuse irradiance wasobtained as the difference of these quantities. The diffuse fraction kdis calculated by

kd 1 0:09kt for kt40:22 65

kd 0:95110:1604kt 4:388k2t 16:638k

3t 12:336k

4t for 0:22< kt40:8

66

kd 0:165 for kt >0:8 67

6.4. Spencer model [21]

Spencer [21] studied the latitude dependence on the mean daily diffuse radiation

with data from 5 stations in Australia (2045S latitude). The following correlation

is proposed

kd a3b3kt for 0:354kt40:75 68

It is presumed that kd has constant values beyond the above range of kt. The

coefficients a3andb3are latitude dependent.

a3 0:940:0118j j 69


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b3 1:1850:0135j j 70

where (degrees) is the latitude. There would be an increasing proportion of diffuse

radiation at higher latitudes due to the increase in average air mass.

6.5. Reindl et al. [19]

Reindl et al. [19] estimated the diffuse fraction kd using two different models

developed with measurements of global and diffuse irradiance on a horizontal sur-

face registered at 5 locations in the USA and Europe (2860N latitude). The first

model (denoted as Reindl-1) estimates the diffuse fraction using the clearness index

as input data and the diffuse fraction kdis given by

kd 1:020:248kt for kt40:3 71

kd 1:451:67kt for 0:3< kt

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6.7. Skartveit and Olseth model [18]

Skartveit and Olseth [18] showed that the diffuse fraction depends also on other

parameters such as solar elevation, temperature and relative humidity. Similararguments were found in the literature [19,3739].

Skartveit and Olseth [18] estimated the direct irradiance Ib from the global irrad-

iance Gt and from the solar elevation angle for Bergen, Norway (60.4N latitude)

with the following equation

Ib Gt 1

sin80

where is a function ofkt and the solar elevation (degrees). The model was vali-

dated with data collected in Aas, Norway (59.7N latitude), Vancouver, Canada

(49.3N latitude) and 10 other stations worldwide. Details of this function are given

below:

Ifkt 1:09c2

1 1:09c21

kt89


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where

1 1d1 d2c03

1

2

1d2 c03

2 90

c03 0:5 1sin c04

d3 0:5

91

c04 1:09c2 c1 92

The constants have to be adjusted for conditions deviating from the validation

domain.

6.8. Maxwell model [23]

A quasi-physical model for converting hourly global horizontal to direct normal

insolation proposed by Maxwell in 1987, was reviewed by Batlles et al. (23). The

model combines a clear physical model with experimental fits for other conditions.

The direct irradiance Ibis given by:

Ib Io d4d5emad6

93

where Io is the extraterrestrial irradiance, is a function of the air mass ma

(dimensionless) and is given by

0:8660:122ma 0:0121m2a 0:000653m

3a 0:000014m

4a 94

and d4, d5and d6are functions of the clearness index kt as given below:

Ifkt40:6,

d4 0:5121:56 kt 2:286 k2t 2:222 k

3t

d5 0:370:962 kt

d6 0:280:923kt2:048 k2t 95

Ifkt >0:6,

d4 5:74321:77kt27:49k2t 11:56k

3t

d5 41:4118:5kt 66:05k2t 31:9k

3t

d6 47:01184:2kt222k2t 73:81k

3t 96

6.9. Louche et al. model [20]

Louche et al. [20] used the clearness indexktto estimate the transmittance of beam

radiation kb. The correlation is given by


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kb 10:627k5t 15:307k

4t 5:205k

3t 0:994k

2t 0:059kt 0:002 97

The correlation includes global and direct irradiance data for Ajaccio (Corsica,

France, 44.9N latitude) between October 1983 and June 1985.Other researchers [40,41] estimated the direct irradiance by means ofkbkt cor-

relations. They found that the solar elevation is an important variable in this type of

correlation. The direct irradiance is then estimated with the following definition of

direct transmittance [23]

Ib kbIo 98

6.10. Vignola and McDaniels model [40]

Vignola and McDaniels [40] studied the daily, 10-day and monthly average beam-

global correlations for 7 sites in Oregon and Idaho, USA (3846N latitude). The

beam-global correlations vary with time of year in a manner similar to the seasonal

variations exhibited by diffuse-global correlations. The direct transmittance kb for

daily correlation is given by

kb 0:0220:28kt 0:828k2t 0:765k

3t for kt50:175 99

kb 0:016kt for kt

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It Ib sin 103

where is a fraction of the diffuse irradiance divided by the direct irradiance and

would be a constant throughout a particular clear-day. The fractionat a given dayN(dimensionless) of a year is given by

N 0:0936 0:041sin N104:5

167

0:004773sin

N24:4

83:5

104

Ib is given by

Ib IoeBma Ioe

B 1sin 0:678

105

where ma is the air mass, and B is the overall atmospheric coefficient of extinction

which varies over the year. The exponent 0.678 compensates for the actual curvature

of the atmosphere.

B N is given by

B N 0:3917397 5:596102sin 2

365N

5:293 103cos

2

365N

1:3594102sin 4

365N

4:0383103cos

4

365N

106

7. Comparison of decomposition models

The kakt correlations of Liu and Jordan [15], Orgill and Hollands [16], Erbs et

al. [17], Reindl-1 [19], Spencer [21] and Lam and Li [22] are plotted in Fig. 5. The

correlations of Orgill and Hollands [16], Erbs et al. [17], Reindl-1 [19] and Lam and

Li [22] are almost identical. The predicted diffuse fraction for Hong Kong by Lam

and Lis correlations is higher than by other correlations for clear-sky conditions

(kt >0:7). The Spencer [21] correlation plotted for yields consistently lowerresults.

The kdkt correlations of Reindl-2 [19] and Skartveit and Olseth [18] were plot-

ted in Fig. 6 for solar elevations of 10, 40 and 90. The results agree well with each

other for kt40:3 and agree reasonably well for 0:3< kt 0:8 and for a lower solar angle 10. The predicted diffuse

fraction for Hong Kong [22] was plotted in the figure for comparison. The predicted

results agree with those of the other two models.

The kbkt correlations proposed by Louche et al. [20], Vignola and McDaniels

[40] and Maxwell [23] with air masses of 1, 2 and 4 are plotted in Fig. 7. The results

of Louche et al. [20] and Vignola and McDaniels [40] are almost identical. TheMaxwell modelled results agree well with the predictions of the other two models. The

kb kt correlation for Hong Kong would be derived from the kdkt correlation


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[Eqs. (77) to (79)] proposed by Lam and Li [22]. With Eqs. (54), (55) and (57), kbcan

be expressed as

kb 1kd kt 107

The derivedkbkt correlation for Hong Kong agrees well withkbpredicted by theothers for kt 0:7 as shown in Fig. 7.

8. Comparison of decomposition models and parametric model

Batlles et al. [23] estimated the hourly values of direct irradiance using the

decomposition models [1620]. The results of these models were compared with

those provided by an atmospheric transmittance parametric model [8] and with the

measured direct irradiances for 6 stations (36.843N latitude, 1.75.4W longitude)

reported by Batlles et al. [43].The aerosol-scattering transmittance adopted by Batlles et al. [23] is calculated

from the visibility.

Fig. 5. Modelled hourly diffuse fractionkdas a function of clearness index kt.


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Fig. 6. Modelled hourly diffuse fraction kdas a function of clearness index kt for various values of the solar

elevation.

Fig. 7. Modelled beam radiation indexkbas a function of clearness indexkt.


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a 0:971:265V0:66is

m0:9a 108

where visibility Vis (km) [2] is given by

Vis 147:9941740:523 1 21

2 0:171 0:011758 0:5h i

109

and

0:552 110

where1 and2 are the A ngstro m turbidity-parameters: 2 is taken to be 1.3 and 1is calculated using the Linke turbidity-factor

TL 85

39:5ew0

47:4 0:1

160:22w0 1 111

where is the solar elevation in degrees and w0 is the precipitable water-content in

cm. The Linke turbidity-factorTLis defined as the number of Rayleigh atmospheres

(an atmosphere clear of aerosols and without water vapour) required to produce a

determined attenuation of direct radiation and is expressed by

TL lRoma 1ln

Io

Ib112

wherelRo(dimensionless) is the Rayleigh optical-thickness. In the study by Batlles et

al. [23], hourly and monthly average hourly values of1 were used.

The decompositions are good for high values of the clearness index (i.e. cloudless

skies and high solar elevations). In such conditions, the model proposed by Louche

et al. [20] estimated the direct irradiance with a 10% root-mean-square (RMS) error

and the estimated direct irradiance by the Iqbal model [8] using hourly values of

has a 4% RMS error. Both models have no significant mean bias error. Batlles et al.[23] concluded that the parametric model is the best when precise information of the

turbidity coefficient is available. However, if there is no turbidity information

available, decomposition models are a good choice.

The clearness index ktand diffuse fraction kdfor Hong Kong (22.3N latitude) were

estimated with the parametric model C [8] as this model offers accurate prediction

over a wide range of conditions [13]. The aerosols, Rayleigh, ozone, gas and water

scattering transmittances a, r, o, g and w respectively, were determined by Eqs.

(108) and (4)(7). The A ngstro m turbidity parameter 2 is taken to be 1.3 and 1 is

taken to be 0, 0.1, 0.2 and 0.4 for the clean, clear, turbid and very turbid atmo-

spheres respectively [8]. The corresponding visibility Vis in Eq. (109) is 337, 28, 11and 5 km, respectively. The measured vertical ozone-layer thickness loz for the

Northern Hemisphere [44] was used to determine the ozone relative optical-path


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length U3 in Eq. (10). The air mass at standard pressure, mr, is approximated by

Kastens formula [8]

mr cosz 0:15 93:885z 1:253

1 113

The precipitable water-vapour thicknessw0 is approximated by Wons equation [8]

w0 0:1e2:25720:05454Tdew 114

where Tdew is the dew-point temperature (C). The monthly average daily tempera-

ture Tand monthly average daily dew-point temperature Tdew recorded between

1961 and 1990 in Hong Kong [33] were adopted in the calculations. The local air

pressure p is taken to be 1013.25 mbar. The ground albedo g is taken to be 0.09,

0.35 and 0.74 for black concrete, uncoloured concrete and white glazed surfaces

respectively (as Hong Kong is a developed city of many high-rise concrete buildings

with curtain walls).

The extraterrestrial radiationI:o was determined by [8]

I:o I

:scEocosz 115

The clearness index kt and the diffuse fraction kd were determined with the mod-

elled global solar irradiance I:t and total diffuse irradiance I

:d. The average values of

the predicted diffuse fraction by the Iqbal model C [8] for clean, clear, turbid andvery turbid atmospheres at ground albedo of 0.09, 0.35 and 0.74 are plotted in Fig. 8.

The predicted kd by Lam and Lis correlations [22] for Hong Kong is shown for

comparison. Lam and Lis results agree with the modeled results for Vis of about 28

km forkt between 0.4 and 0.7 and fall in the range ofVisbetween 11 and 28 km for

ktbetween 0.2 and 0.4.

9. Other studies of solar radiation models

Literature results [37,38] showed that the diffuse fraction depends also on othervariables like atmospheric turbidity, surface albedo and atmospheric precipitable

water.

Chendo and Maduekwe [39] studied the influence of four climatic parameters on the

hourly diffuse fraction in Lagos, Nigeria. It was found that the diffuse fraction of the

global solar-radiation depended on the clearness index kt, the ambient temperature,

the relative humidity and the solar altitude. The standard error of the Liu and Jordan-

type [16,17,19] was reduced by 12.8% when the solar elevation, ambient temperature

and relative humidity were included as predictor variables for the 2-year data set.

Tests with existing correlations showed that the Orgill and Hollands model [16]

performed better than the other two models [17,19].Elagib et al. [45] proposed a model to compute the monthly average daily global

solar-radiation Gt (MJ m2) for Bahrain (22N latitude) from commonly measured


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meteorological parameters. The parameters are monthly average daily relative-

humidity RH (%), the monthly average daily temperature-range T (C), the

monthly average daily temperatureT(C), the monthly average daily hours of sun-

shine S (h) and the monthly average daily extraterrestrial solar-radiation Go (MJ

m2).

For JanuaryJune, Gt can be computed by one of the following correlation equations:

Gt 70:83850:7886RH 116

Gt 55:85280:6440 RHS 117

Gt 49:26370:6751 RHTS 118

For JulyDecember,

Gt 4:5456e0:0453T 119

Fig. 8. Modelled hourly diffuse fractionkdas a function of clearness indexkt for Hong Kong.


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Gt 1:53580:6078Go 120

Gt 34:55770:4642 RHT 121

Gt 18:94690:2663 RHTGo 122

The 3-variable models yielded the smallest errors yet the 1-variable models can

give close estimations.

Tovar et al. [46] studied the probability density distributions of 1-min values of

global irradiance, conditioned to the optical air-mass, considering them to be an

approximation to the instantaneous distributions. It was found that the bimodality that

characterizes these distributions increases with optical air-mass. The function based on

Boltzmanns statistics can be used for the generation of synthetic radiation data.

Expressing the distribution as a sum of two functions provides an appropriate mod-

elling of the bimodality feature that can be associated with the existing two levels of

irradiance corresponding to two extreme atmospheric situations the cloudless and

cloudy conditions.

In addition to the shortwave (0.32.0 mm) radiation from the Sun, the Earth

receives longwave radiation (4100 mm) from the atmosphere. The air conditions at

ground level largely determine the magnitude of the incoming radiation [5] and the

downward radiation from the atmosphere qratcould be expressed as

qrat "atT4at 123

where "at (dimensionless) is the atmosphere emittance, is the StefanBoltzmann

constant (5.67108 W m2 K4) andTat (K) is the ground-level temperature.

The apparent sky-temperature Tsky(K) is defined as the temperature at which the

sky (as a blackbody) emits radiation at the rate actually emitted by the atmosphere

at ground-level temperature with its actual emittance "at. Then,

T4sky "atT4at 124

or

T4sky "atT4at 125

The atmosphere emittance is a complex function of air temperature and moisture

content [47]. A simple relationship [5], which ignores the vapour pressure of the

atmosphere could be used to estimate the apparent sky-temperature

Tsky 0:0552T1:5at 126


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10. Conclusion

Solar radiation models are desirable for designing solar-energy systems and good

evaluations of thermal environments in buildings. Two parametric models (Iqbalmodel C [8] and ASHRAE model [5]) were used to predict the beam, diffuse and

global irradiances under clear-sky conditions for Hong Kong. At zero zenith angle,

a maximum difference of beam irradiance of not more than 7% is obtained between

the 2 models. The beam irradiance deviation of less than 8% was found for Sep-

temberApril, and less than 12% for MayAugust when the zenith angle is less than

60. The deviation of the global irradiance predicted between the 2 models was only

5%. A large deviation, up to 41%, was found between the predicted values of the

diffuse irradiance with the 2 models. The ASHRAE model would be less accurate

for diffuse radiation predictions because the ground albedo and aerosol-generated

diffuse irradiance are not included.

Sunshine hours and cloud-cover data are available easily for many stations: cor-

relations for predicting the insolation with sunshine hours are reviewed. Seven

models using the A ngstro m-Prescott equation to predict the average daily global

radiation with hours of sunshine have been reviewed. Estimation of monthly aver-

age hourly global radiation was discussed. The 7 models were applied to predict the

daily global radiation for Hong Kong and comparison with measured data was

made. Predictions made by Bahel et al. [29], Alnaser [30] and Louche et al. [20] agree

with the measured data, while the other models provide reasonable predictions.

Values of global and diffuse radiation for individual hours are essential for manyresearch and engineering applications. Twelve decomposition models in the litera-

ture commonly used to predict the diffuse component using the measured hourly

global data are reviewed. The models were used to predict the hourly diffuse fraction

for Hong Kong and compared with the model based on Hong Kong measurements

[22]. Good agreement was found among the model predictions, but lower diffuse

fractions were predicted by some models [1517,19,21] for clear-sky conditions. The

beam-radiation index for Hong Kong was estimated by the Louche et al. [20], Vig-

nola and McDaniels [40] and Maxwell [23] models and compared with the prediction

of Lam and Lis model [22]. Again, good agreement was found except that a lower

beam-radiation index was predicted by Lam and Lis model [22] under clear-skyconditions.

The clearness index kt and diffuse fraction kd for Hong Kong were also deter-

mined with the Iqbal model C [8] for 4 atmospheric conditions (clear, clean, turbid

and very turbid atmospheres) for 3 cases of ground albedo (corresponding to black

concrete, uncoloured concrete and white glazed surface). The results are then com-

pared with those predicted by Lam and Lis model [22]. It was found that Lam and

Lis predictions agree with the modeled results for Vis about 28 km for kt between

0.4 and 0.7, and fall for Vis between 11 and 28 km for kt between 0.2 and 0.4. The

parametric model would provide accurate predictions of solar radiation as turbidity

information is included and good evaluations of thermal environments in buildingsare therefore possible. If precise atmospheric information, such as optical properties

of clouds, cloud amount, thickness, position, the number of layers and turbidity, is


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not available, decomposition models based on measured hourly global radiation

would be a good choice.

References

[1] Lu Z, Piedrahita RH, Neto CDS. Generation of daily and hourly solar radiation values for modeling

water quality in aquaculture ponds. Transactions of the ASAE 1998;41(6):18539.

[2] Machler MA, Iqbal M. A modification of the ASHRAE clear-sky irradiation model. ASHRAE

Transactions 1985;91(1a):10615.

[3] Cartwright TJ. Here comes the Sun: solar energy from a flat-plate collector. In: Modeling the world

in a spreadsheet-environmental simulation on a microcomputer. London: The Johns Hopkins Uni-

versity Press, 1993. p. 12144.

[4] Trujillo JHS. Solar performance and shadow behaviour in buildings case study with computer

modelling of a building in Loranca, Spain. Building and Environment 1998;33(2-3):11730.[5] ASHRAE handbook: HVAC applications. Atlanta (GA): ASHRAE, 1999.

[6] Li DHW, Lam JC. Solar heat gain factors and the implications for building designs in subtropical

regions. Energy and Buildingsz 2000;32(1):4755.

[7] Iqbal M. Estimation of the monthly average of the diffuse component of total insolation on a hor-

izontal surface. Solar Energy 1978;20(1):1015.

[8] Iqbal M. An introduction to solar radiation. Toronto: Academic press, 1983.

[9] Davies JA, McKay DC. Estimating solar irradiance and components. Solar Energy 1982;29(1):5564.

[10] Sherry JE, Justus CG. A simple hourly all-sky solar-radiation model based on meteorological para-

meters. Solar Energy 1984;32(2):195204.

[11] Rao CRN, Bradley WA, Lee TY. The diffuse component of the daily global solar-irradiation at

Corvallis, Oregon (USA). Solar Energy 1984;32(5):63741.

[12] Carroll JJ. Global transmissivity and diffuse fraction of solar radiation for clear and cloudy skies asmeasured and as predicted by bulk transmissivity models. Solar Energy 1985;35(2):10518.

[13] Gueymard C. Critical analysis and performance assessment of clear-sky solar-irradiance models

using theoretical and measured data. Solar Energy 1993;51(2):12138.

[14] Gueymard C. Mathematically integrable parametrization of clear-sky beam and global irradiances

and its use in daily irradiation applications. Solar Energy 1993;50(5):38597.

[15] Liu BYH, Jordan R C. The inter-relationship and characteristic distribution of direct, diffuse and

total solar radiation. Solar Energy 1960;4(3):119.

[16] Orgill JF, Hollands KGT. Correlation equation for hourly diffuse radiation on a horizontal surface.

Solar Energy 1977;19(4):3579.

[17] Erbs DG, Klein SA, Duffie JA. Estimation of the diffuse radiation fraction for hourly, daily and

monthly-average global radiation. Solar Energy 1982;28(4):293302.

[18] Skartveit A, Olseth JA. A model for the diffuse fraction of hourly global radiation. Solar Energy

1987;38(4):2714.

[19] Reindl DT, Beckman WA, Duffie JA. Diffuse fraction corrections. Solar Energy 1990;45(1):17.

[20] Louche A, Notton G, Poggi P, Simonnot G. Correlations for direct normal and global horizontal

irradiations on a French Mediterranean site. Solar Energy 1991;46(4):2616.

[21] Spencer JW. A comparison of methods for estimating hourly diffuse solar-radiation from global

solar-radiation. Solar Energy 1982;29(1):1932.

[22] Lam JC, Li DHW. Correlation between global solar-radiation and its direct and diffuse components.

Building and Environment 1996;31(6):52735.

[23] Batlles FJ, Rubio MA, Tovar J, Olmo FJ, Alados-Arboledas L. Empirical modeling of hourly direct

irradiance by means of hourly global irradiance. Energy 2000;25(7):67588.

[24] Beckman WA, Bugler JW, Cooper PI, Duffie JA, Dunkle RV, Glaser PE et al. Units and symbols in

solar energy. Solar Energy 1978;21(1):658.

[25] A ngstro m A. Solar and terrestrial radiation. Quarterly Journal of Royal Meteorological Society

1924;50:1216.


8/13/2019 jhaa1

34/34

[26] Gueymard C, Jindra P, Estrada-Cajigal V. A critical look at recent interpretations of the A ngstro m

approach and its future in global solar radiation prediction. Solar Energy 1995;54(5):35763.

[27] Rietveld MR. A new method for estimating the regression coefficients in the formula relating solar

radiation to sunshine. Agricultural Meteorology 1978;19:243352.[28] Gopinathan KK. A general formula for computing the coefficients of the correction connecting glo-

bal solar-radiation to sunshine duration. Solar Energy 1988;41(6):499502.

[29] Bahel V, Srinivasan R, Bakhsh H. Solar radiation for Dhahran, Saudi Arabia. Energy

1986;11(10):9859.

[30] Alnaser WE. Empirical correlation for total and diffuse radiation in Bahrain. Energy 1989;14(7):409

14.

[31] Srivastava SK, Singh OP, Pandey GN. Estimation of global solar-radiation in Uttar Pradesh (India)

and comparison of some existing correlations. Solar Energy 1993;51(1):279.

[32] Sen Z. Fuzzy algorithm for estimation of solar irradiation from sunshine duration. Solar Energy

1998;63(1):3949.

[33] Hong Kong Observatory, Monthly meteorological normals (19611990) and extremes (18841939,

19471999) for Hong Kong. http://www.info.gov.hk/hko/wxinfo/climat/climato.htm, 2000.[34] Hay JE. Calculation of monthly mean solar-radiation for horizontal and inclined surfaces. Solar

Energy 1979;23(4):3017.

[35] Mohandes M, Balghonaim A, Kassas M, Rehman S, Halawani TO. Use of radial basis functions for

estimating monthly mean daily solar-radiation. Solar Energy 2000;68(2):1618.

[36] Collares-Pereira M, Rabl A. The average distribution of solar radiation-correlations between diffuse

and hemispherical and between daily and hourly insolation values. Solar Energy 1979;22(2):15564.

[37] Garrison JD. A study of the division of global irradiance into direct and diffuse irradiances at thirty-

three US sites. Solar Energy 1985;35(4):34151.

[38] Camps J, Soler MR. Estimation of diffuse solar-irradiance on a horizontal surface for cloudless days:

a new approach. Solar Energy 1992;49(1):5363.

[39] Chendo MAC, Maduekwe AAL. Hourly global and diffuse radiation of Lagos, Nigeria-correlation

with some atmospheric parameters. Solar Energy 1994;52(3):24751.[40] Vignola F, McDaniels DK. Beam-global correlations in the Northwest Pacific. Solar Energy

1986;36(5):40918.

[41] Jeter SM, Balaras CA. Development of improved solar radiation models for predicting beam trans-

mittance. Solar Energy 1990;44(3):14956.

[42] Al-Riahi M, Al-Jumaily KJ, Ali HZ. Modeling clear-weather day solar-irradiance in Baghdad, Iraq.

Energy Conversion Management 1998;39(12):128994.

[43] Batlles FJ, Olmo FJ, Alados-Arboledas L. On shadowband correction methods for diffuse irradiance

measurements. Solar Energy 1995;54(2):10514.

[44] Robinson N. Solar radiation. New York: American Elsevier, 1966.

[45] Elagib NA, Babiker SF, Alvi SH. New empirical models for global solar radiation over Bahrain.

Energy Conversion Management 1998;39(8):82735.

[46] Tovar J, Olmo FJ, Alados-Arboledas L. One-minute global irradiance probability density distribu-

tions conditioned to the optical air-mass. Solar Energy 1998;62(6):38793. (1998).

[47] Reitan CH. Surface dew-point and water-vapor aloft. Journal of Applied Meteorology

1963;2(6):7769.


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